Coupled Bending Research Articles

Study on the Coupled Bending and Torsional Vibrations of Turbo-Generator Unit with Rub-Impact

An increment transfer matrix equations based on step-by-step integration method and traditional transfer matrix method are deduced Combined with multi-mass model and Riccati method, the increment transfer matrix method is put forward, that can be directly used to analyze the dynamic response of the coupled bending and torsional vibrations of turbo-generator shafts with rub-impact. Taking a turbo-generator unit as example, the vibration character of rub-impact fault is analyzed when unit starts up. The research results show that rubbing will make vibration amplitude increase when the rotational speed is lower than the first critical speed; however, when the speed is higher than the first critical speed, rubbing will make the rotor mass unbalance reduce, thereby vibration amplitude will reduce slightly.

An Increment Transfer Matrix Method for the Coupled Bending and Torsional Vibrations of Turbo-Generator Shafts with Rub-Impact

An Increment Transfer Matrix Method for the Coupled Bending and Torsional Vibrations of Turbo-Generator Shafts with Rub-Impact

Analysis of Characteristics of Coupled Bending Vibrators Used as a Force Sensor

Analysis of Characteristics of Coupled Bending Vibrators Used as a Force Sensor

Analysis of Characteristics of Coupled Bending Vibrators Used as a Force Sensor

The characteristics of the coupled bending vibrator that linearly connects the two bending vibrators are analyzed by the finite-element method, and then the characteristics of the coupled bending vibrator used as a force sensor are investigated. The coupled bending vibrator is applied to an acceleration sensor as a force sensor. The vibration mode in which the two vibrators of the coupled vibrator are out of phase is used here and is very stable because it does not couple with other vibration modes. When the axial force is applied, the amplitude of one vibrator of the coupled vibrator decreases with increase in compressive force. The amplitude of the other vibrator also decreases when the force is applied in the opposite direction. Therefore, the coupled vibrator is used as a force sensor that can estimate the axial force as a change in the amplitude of the vibrator.

The Analysing of Coupled Bending Tosion and Shear-Lag of Curved Box Girder Bridge Considering Prestress and Initial Curvature

Based on the theory of thin-walled curved bar and considered the impact of initial curvature and prestress on the vertical flexure, the shear lag warp displacement function is replenished on the basic deformation of the curved box girder flange plate, for which the longitudinal dispersive function is utilized. According to the energy functional differential methods, the coupled bending tosion and shear lag of elasticity governing differential equations of curved prestressed box girder are deduced with different boundary conditions in considering prestress and initial curvature. The numerical solution is gained by Galerlein method. The calculated value of this article coincides well with the value of the experiment and finite-element method. It builds the theory analysing basic of shear lag effect of curved box girder bridge considering prestress and initial curvature.

The Analysing of Coupled Bending Tosion and Shear-Lag of Curved Box Girder Bridge Considering Prestress and Initial Curvature

Based on the theory of thin-walled curved bar and considered the impact of initial curvature and prestress on the vertical flexure, the shear lag warp displacement function is replenished on the basic deformation of the curved box girder flange plate, for which the longitudinal dispersive function is utilized. According to the energy functional differential methods, the coupled bending tosion and shear lag of elasticity governing differential equations of curved prestressed box girder are deduced with different boundary conditions in considering prestress and initial curvature. The numerical solution is gained by Galerlein method. The calculated value of this article coincides well with the value of the experiment and finite-element method. It builds the theory analysing basic of shear lag effect of curved box girder bridge considering prestress and initial curvature.

Reduced Modeling for Turbine Rotor-Blade Coupled Bending Vibration Analysis

In a traditional turbine-generator set, rotor shaft designers and blade designers have their own models and design process which neglects the coupled effect. Since longer blade systems have recently been employed (Saito et al. 1998, “Development of a 3000 rpm 43-in. last stage blade with high efficiency and reliability,” International Joint Power Generation Conference, pp. 89–96.) for advanced turbine sets to get higher output and efficiency, additional consideration is required concerning rotor bending vibrations coupled with a one-nodal (k = 1) blade system. Rotor-blade coupled bending conditions generally include two types so that the parallel and tilting modes of the shaft vibrations are respectively coupled with in-plane and out-of-plane modes of blade vibrations with a one-nodal diameter (k = 1). This paper proposes a method to calculate the natural frequency of a shaft blade coupled system. According to this modeling technique, a certain blade mode is reduced to a single mass system, which is connected to the displacement and angle motions of the shaft. The former motion is modeled by the m-k system to be equivalent to the blade on the rotating coordinate. The latter motion is commonly modeled in discrete form using the beam FEM on an inertia coordinate. Eigenvalues of the hybrid system covering both coordinates provide the natural frequency of the coupled system. In order to solve the eigenfrequencies of the coupled system, a tracking solver method based on sliding mode control concept is used. An eight-blade system attached to a cantilever bar is used for an example to calculate a coupled vibration with a one-nodal diameter between the blade and shaft.

3Pb3-5 Analysis of Characteristics of Coupled Bending Vibrators Used as a Force Sensor(Poster Session)

3Pb3-5 Analysis of Characteristics of Coupled Bending Vibrators Used as a Force Sensor(Poster Session)

535 タービンの翼軸連成曲げ振動解析向け等価質点モデル化

535 タービンの翼軸連成曲げ振動解析向け等価質点モデル化

Resonance and Instability of Blade-Shaft Coupled Bending Vibrations with In-plane Blade Vibration

Abstract As a major component of a power plant, a turbine generator must have sufficient reliability. Longer blades have lower natural frequency, thereby requiring that the design of the shaft and blade takes into account the coupling of the blade vibration mode, nodal diameter k =0 and k =1 with vibration of the shaft. The present work analyzes the coupling of the translation motion of the shaft with in-plane vibration of the blades with k =1 modes. At a rotational speed Ω 1 =|ω s −ω b |, the resonance of the blades has a relatively large amplitude. A violent coupled resonance was observed at a rotational speed Ω 2 =ω s +ω b . Resonance in blade vibration at Ω 1 =|ω s −ω b | was experimentally confirmed. Keywords : Blade-Shaft coupled vibration, Bending vibration, Resonance, Instability, In-plane vibration, Turbine generator 1. Introduction As a major component of the power plant, a turbine generator must have sufficient reliability. The rotary shaft system is usually designed to give a low Q-factor and avoid resonance based on the rotor-dynamic analysis of the bending and torsional vibration of the shaft-bearing system, in which the blades are assumed to be a rigid disk. Meanwhile, the blade is designed to avoid resonance under the rated operation conditions based on analysis of a single blade under ideal conditions where the shaft is assumed to be fixed [1],[2]. The long blades used in turbine generators for nuclear power plants have low natural frequencies, thereby necessitating a design that takes into consideration blade-shaft coupled vibration. General conditions for coupling of blade and shaft vibrations are shown in Table 1. Blade

Effects of Geometric and Structural Parameters on Coupled Bending Torsion Flutter in Turbo Machinery Blades

Effects of Geometric and Structural Parameters on Coupled Bending Torsion Flutter in Turbo Machinery Blades

Effects of Geometric and Structural Parameters on Coupled Bending Torsion Flutter in Turbo Machinery Blades

Effects of Geometric and Structural Parameters on Coupled Bending Torsion Flutter in Turbo Machinery Blades

Resonance and Instability of Blade-Shaft Coupled Bending Vibrations

Resonance and Instability of Blade-Shaft Coupled Bending Vibrations

Coupled lateral bending–torsional vibration sensitivity of atomic force microscope cantilever

We study the influence of the contact stiffness and the ration between cantilever and tip lengths on the resonance frequencies and sensitivities of lateral cantilever modes. We derive expressions to determine both the effective resonance frequency and the mode sensitivity of an atomic force microscope (AFM) rectangular cantilever. Once the contact stiffness is given, the resonance frequency and the sensitivity of the vibration modes can be obtained from the expression. The results show that each mode has a different resonant frequency to variations in contact stiffness and each frequency increased until it eventually reached a constant value at very high contact stiffness. The low-order vibration modes are more sensitive to vibration than the high-order mode when the contact stiffness is low. However, the situation is reversed when the lateral contact stiffness became higher. Furthermore, increasing the ratio of tip length to cantilever length increases the vibration frequency and the sensitivity of AFM cantilever.

A quadrilateral element based on refined global-local higher-order theory for coupling bending and extension thermo-elastic multilayered plates

In this paper a refined higher-order global-local theory is presented to analyze the laminated plates coupled bending and extension under thermo-mechanical loading. The in-plane displacement fields are composed of a third-order polynomial of global coordinate z in the thickness direction and 1,2–3 order power series of local coordinate ζ k in the thickness direction of each layer, which is identical to the 1,2–3 global-local higher-order theory by Li and Liu [Li, X.Y., Liu, D., 1997. Generalized laminate theories based on double superposition hypothesis. Int. J. Numer. Methods Eng. 40, 1197–1212] Moreover, a second-order polynomial of global coordinate z in the thickness direction is chosen as transverse displacement field. The transverse shear stresses can satisfy continuity at interfaces, and the number of unknowns does not depend on the layer numbers of the laminate. Based on this theory, a quadrilateral laminated plate element satisfying the requirement of C 1 continuity is presented. By solving both bending and thermal expansion problems of laminates, it can be found that the present refined theory is very accurate and obviously superior to the existing 1,2–3 global-local higher-order theory. The most attractive feature of this theory is that the transverse shear stresses can be accurately predicted from direct use of constitutive equations without any post-processing method. It is also shown that the present quadrilateral element possesses higher accuracy.

Coupled bending and torsional vibration of a beam with in-span and tip attachments

In this paper, the coupled flexural–torsional free and forced vibrations of a beam with tip and/or in-span attachments are studied. First, a mathematical model is established, which consists of a beam with several tip attachments, i.e, a tip mass of non-negligible dimensions, a linear spring grounding the tip mass, and a torsional spring connected at the end of the beam. The modal functions of this model and the orthogonality condition among them are derived. For the purpose of verification the properties of the tip attachments are changed, and the numerical results obtained are compared with those given in the relevant literature. Effects of tip mass and distributed mass in-span on natural frequencies and modes are investigated for two cantilever beams with different cross sections. An application of the orthogonality condition in the case of a beam with tip mass is also presented for a forced vibration example.

Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli–Euler beams

The dynamic transfer matrix is formulated for a straight uniform and axially loaded thin-walled Bernoulli–Euler beam element whose elastic and inertia axes are not coincident by directly solving the governing differential equations of motion of the beam element. Bernoulli–Euler beam theory is used, and the cross section of the beam does not have any symmetrical axes. The bending vibrations in two perpendicular directions are coupled with torsional vibration and the effect of warping stiffness is included. The dynamic transfer matrix method is used for calculation of exact natural frequencies and mode shapes of the nonsymmetrical thin-walled beams. Numerical results are given for a specific example of thin-walled beam under a variety of end conditions, and exact numerical solutions are tabulated for natural frequencies and solutions calculated by the other method are also tabulated for comparison. The effects of axial force and warping stiffness are also discussed.

Corrigendum to ?Coupled bending and torsional vibration of axially loaded Bernoulli?Euler beams including warping effects? 6Applied Acoustics 65 (2004) 153?1709

Corrigendum to ?Coupled bending and torsional vibration of axially loaded Bernoulli?Euler beams including warping effects? 6Applied Acoustics 65 (2004) 153?1709

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.

Coupled bending and torsional vibration of axially loaded Bernoulli–Euler beams including warping effects

The dynamic transfer matrix method for determining natural frequencies and mode shapes of the bending-torsion coupled vibration of axially loaded thin-walled beams with monosymmetrical cross sections is developed by using a general solution of the governing differential equations of motion based on Bernoulli–Euler beam theory. This method takes into account the effect of warping stiffness and gives allowance to the presence of axial force. The dynamic transfer matrix is derived in detail. Two illustrative examples on the application of the present theory are given for bending-torsion coupled beams with thin-walled open cross sections. The effects of axial load and warping stiffness on coupled bending-torsional frequencies are discussed. Compared with those available in the literature, numerical results demonstrate the accuracy and effectiveness of the proposed method.